Study of $B\to \Lambda\bar\Lambda K$ and $B\to \Lambda\bar\Lambda \pi$

We study three-body charmless baryonic B decays of $B \to \Lambda\bar\Lambda P$ with $P=\pi$ and $K$ in the standard model. We find that the branching ratios of the $K$ modes are about one order of magnitude larger than those of the corresponding $\pi$ modes unlike the cases of $B\to p\bar p P$. Explicitly, we obtain that $Br(B^-\to \Lambda\bar \Lambda K^-)=(2.8\pm 0.2)\times 10^{-6}$ and $Br(\bar B^0 \to\Lambda\bar \Lambda \bar K^0)=(2.5\pm 0.3)\times 10^{-6}$. The former agrees well with the BELLE experimental measurement of $(2.91^{+0.90}_{-0.70}\pm 0.38)\times 10^{-6}$, while the latter should be seen at the ongoing B factories soon.

There have been lots of attentions recently on charmless three-body baryonic B decays due to the several new experimental measurements by BELLE and BABAR [1,2,3,4,5].
There are mainly two kinds of approaches to study the baryonic B decays in the literature.
One is the pole model, presented in Refs.[9,10,11], where the intermediate particles couple dominantly to the final states.The other, proposed in Refs.[7,12,13], is based on the QCD counting rules [14,15], which deal with the baryonic form factors by power expansions.We note that global χ 2 fits in Ref. [16] for Br(B − → ppπ − ), Br(B 0 → ppK 0 ) and Br(B − → ppK − ) [1] with the QCD counting rules have been performed and consistent results with data have been derived.Furthermore, various radiative three-body baryonic B decays [16] have been studied.
In this report, we will concentrate on the three-body charmless baryonic decays of based on the QCD counting rules.We note that so far there has been no theoretical study on the modes in Eq. ( 3) in the literature.In particular, we do not know the reason why the decay branching ratio of B − → Λ Λπ − is smaller than that of B − → Λ ΛK − , whereas the corresponding pp modes are comparable, in terms of the data shown in Eqs. ( 1) and ( 2).In addition, these decays are of great interest since they provide opportunities for probing T violating effects [17] due to the measurable polarization of Λ [18].
There are two types of diagrams which contribute to the decays in Eq. (3) under the factorization approximation [7], with Figs.2a and 2b representing typical diagrams of a current-produced baryon pair with B → P and a current-produced P with B → B B, named as C and T terms, respectively.The amplitude of B → B BP with B = p or Λ and P = π or K in the factorization approximation is given by [7] A where [19,20] with q = d(s) for P = π(K), while T (B → B BP ) is decomposed with the P -meson decay constant f P induced from the P creation and B → B B through scalar and pseudoscalar currents as with q ′ = u(d) for charged (neutral) modes, and where 10), defined in Refs.[19,20], and N c is the effective color number.The coefficients a i and c are related by To calculate the decay rate, we need to know the hadronic transition matrix elements in Eqs. ( 5) and (6).The B → P transition matrix element can be parameterized as where t ≡ (p B + pB) 2 and F B→P 1,0 (t) are defined by [22] F B→P with the input parameter values of F B→π For those of the baryon pair involving the vector, axial-vector, scalar and pseudoscalar currents in Eq. ( 5), we have where V µ = qi γ µ q j , A µ = qi γ µ γ 5 q j , S = qi q j and P = qi γ 5 q j with q i = u, d and s.We note that F 2 (t) alone can not be determined by the present experimental data.However, F 2 (t) can be ignored since it acquires one more 1/t than F 1 (t) according to the power expansion in a perturbative QCD re-analysis [23,24].By using equation of motion and adopting zero quark mass limits, we have In terms of the SU(3) flavor symmetry, the form factors [F 1 (t) + F 2 (t)], g A (t), f S (t) and g P (t) in Eq. ( 11) can be related to another set of form factors D X (t), F X (t) and S X (t) where X = V, A, S and P denote vector, axial-vector, scalar, pseudo-scalar currents, defined in Table I, respectively.It is noted that the zero value of pp|(ss) X |0 in the second column is due to the OZI suppression rule.As an illustration, we take Λ Λ|(ūu + dd + ss) V,A |0 in Form Factor (X=V,A,S,P) Eq. ( 5) and we have For D X (t), F X (t), and S X (t), since they are related to the nucleon magnetic (Sachs) form factors G p(n) M (t), we adopt the results for G p(n) M (t) in Ref. [25], which are extracted from experiments.
The functions of G p(n) M (t), D X (t), F X (t) and S X (t) are parameterized as with The input numbers can be found in Refs.[7,12,13].Explicitly, we take γ = 2.148, ), s1 = n q (x 1 − 2y 1 ), s2 = n q s 2 , d2 = f2 = −952 GeV 6 and Λ 0 = 0.3 GeV.We remark that the parameter n q in Eq. ( 15) corresponds to the (m B − mB′)/(m q − m q′ ) term in connecting scalar form factors to vector ones in the case of B = B′ .In B → B BP decays, this parameter is not well-defined.However, by taking m q′ → m q and mB′ → m B , it has been shown [12] that n q is around 1.3 − 1.4.Here we fix n q ≃ 1.4.
For the B → B B transition in Eq. ( 6), we first discuss the case with B = p.For B − → pp, one has [7] where p = p B − (p p + p p ).Note that F S = F P as shown in Ref. [12].For B0 → pp, one gets The form factors in Eqs. ( 16) and ( 17) can be related [7,16] by the SU(3) symmetry and the helicity conservation [14,15] and one obtains that Moreover, the three form factors F A , F V and F P in Eq. ( 16) can be simply presented by [7] where C i (i = A, V, P ) are new parameterized form factors, which are taken to be real.By following the approach of Refs.[7,15], the form factors in B −,0 → Λ Λ transitions are given by where It is interesting to note that F Λ Λ P (S) = 0 from Eqs. (20) and (21).The reason1 for this result is that F Λ Λ P (S) correspond to a helicity (chirality) flipped terms in the B → Λ Λ matrix elements in the large momentum transfer.It is well known that the spin of Λ is carried by the s-quark component.Hence, such a term requires a chirality flip in the s-quark, which is absent in the decay amplitude of B → Λ Λ.
Br(B − → Λpγ) [4] To obtain these unknown form factors, we use the χ 2 fit with the experimental data in Eq. ( 1).Here we have neglected [16] the C P term since it has one more 1/t over C A and C V as shown in Eq. ( 19).Therefore, we keep the numbers of the degree of freedom (ndf) to be 2.In our fit, we also include the uncertainties from the Wolfenstein parameters [26] in the CKM matrix.Explicitly, we use λ = 0.2200 ± 0.0026, A = |V cb |/λ 2 = 0.853 ± 0.037, ρ = 0.20 ± 0.09 and η = 0.33 ± 0.05 [27].We follow the Refs.[20,28] to deal with a i (i = 1, ..., 10) and we take Wilson coefficients from Ref. [29].
The experimental inputs and the fitted results are shown in Table II.With the fitted values in Table II, the theoretical predictions with 1σ error for B − → Λ ΛP are given by which agree very well with the recent BELLE data [5] in Eq. ( 1).We note that the spectrum of B − → Λ ΛK − shown in Fig. 2b is consistent with that in Ref. [5] by BELLE.
Our predicted values in Eq. (22) show that ).This result can be easily understood from the theoretical point of view.As B − → ppπ − can not obtain a large contribution from C in Eq. ( 5) since a 2 is color suppressed, its main contribution is from a 1 term in T as seen in Eqs. ( 6) and ( 7), whereas B − → ppK − is due to the penguin part which gives contributions to the terms in both C and T .However, it is not the case for B − → Λ ΛK − since Λ Λ|(ss) X |0 in Eq. ( 5) escapes from the OZI suppression.Its contribution is mainly from C which is enhanced by a 6 with the chiral enhancement, whereas it is suppressed in T .For B − → Λ Λπ − , on the other hand, because of the a 2 color suppression in C and the small contribution in T , its branching ratio is small.Moreover, to explicitly see the T -suppression in the Λ Λ case, we show the spectra of B − → ppπ − and B − → Λ Λπ − in Fig. 2. We note that the two modes contain the same T -amplitudes in Eq. ( 6).As shown in Fig. 2 the main contributions to the rates are due to the threshold effect, resulting from the form factors of Similarly, we can also study the neutral decay modes of B0 → Λ Λ K0 (π 0 ), which have not been measured yet.The decay branching ratios are found to be As seen from Eq. ( 23), the decay branching ratio of B0 → Λ Λ K0 is almost the same as that of B − → Λ ΛK − , whereas B0 → Λ Λπ 0 is still suppressed as B − → Λ Λπ − .We note that the errors of Br(B − → Λ ΛK − ) and Br( B0 → Λ Λ K0 ) in Eqs. ( 22) and ( 23) are small since the main contributions are not from C Λ Λ A,V which receive almost all uncertainties from the data.To describe the possible non-factorizable effects, we also fit the data with N c = 2 and ∞.
As expected, we find that the branching ratios of the K modes are slightly changed, while the central values of B − → Λ Λπ − and B0 → Λ Λπ 0 shift to 2.3 and 1.3 (0.7 and 0.1) for N c = 2 (∞), respectively.We may conclude that the two π modes remain small even with including all possible non-factorizable effects.However, as pointed in Ref. [30], a 2 can only be determined by the experimental data in the two-body B decays, since the experimental data of Br( B0 → π 0 π 0 ) and Br( B0 → D 0 π 0 ) are much larger than the theoretical values, which means the failure of the factorization approximation.On the other hand, in the threebody baryonic B decays, due to the complicated topology of Feynman diagrams, it is not as easy as those of the two-body decays to influence a 2 by the annihilations [31] as well as the final state interactions [32].Therefore, the value of a 2 may not change much.Nevertheless, we leave the surprise to the experimentalists if the factorization does not work well in these two modes as those in the two-body decays. In

FIG. 1 :
FIG. 1: Diagrams for B → B BP with B = p and Λ, P = π(K) for q = d(s) and q ′ = u(d) for charged (neutral) modes, and where (a) and (b) represent C and T terms, respectively.

FIG. 2 :
FIG. 2: dBr/dm B B as a function of m B B for (a) B − → ppP and (b) B − → Λ ΛP , where the solid and dashed lines for P = K − and P = π − , respectively.
Furthermore, the relations from the numerator of the form factor are C Λ Λ A ≃ C A /2 and C Λ Λ V ≃ −C V /10.Once we combine the relations above and integrate them through the phase space, we obtain that Br(B − → Λ Λπ − ) ≃ O(10 −1 )Br(B − → ppπ − ).

TABLE II :
Fits of (C A ,C V ) in units of GeV 4 .